Kyungpook Mathematical Journal

Department of Mathematics, Kyungpook National University (경북대학교 자연과학대학 수학과)
권호 표지
DOI QR코드

DOI QR Code

Evolution of the First Eigenvalue of Weighted p-Laplacian along the Yamabe Flow

  • Azami, Shahroud (Department of Mathematics, Faculty of Sciences, Imam Khomeini International University)
  • Received : 2018.07.14
  • Accepted : 2019.01.21
  • Published : 2019.06.23
Download PDF
⟨ Previous Next ⟩

Abstract

Let M be an n-dimensional closed Riemannian manifold with metric g, $d{\mu}=e^{-{\phi}(x)}d{\nu}$ be the weighted measure and ${\Delta}_{p,{\phi}}$ be the weighted p-Laplacian. In this article we will study the evolution and monotonicity for the first nonzero eigenvalue problem of the weighted p-Laplace operator acting on the space of functions along the Yamabe flow on closed Riemannian manifolds. We find the first variation formula of it along the Yamabe flow. We obtain various monotonic quantities and give an example.

Keywords

References

  1. S. Azami, Eigenvalue variation of the p-Laplacian under the Yamabe flow, Cogent math., 3(2016), Art. ID 1236566, 10 pp.
  2. S. Azami,Monotonicity of eigenvalues of Witten-Laplace operator along the Ricci-Bourguignon flow, AIMS mathematics, 2(2)(2017), 230-243. https://doi.org/10.3934/Math.2017.2.230
  3. X. Cao, Eigenvalues of ($-{\Delta}+\frac{R}{2}$) on mannifolds with nonegative curvature operator, Math. Ann., 337(2)(2007), 435-441. https://doi.org/10.1007/s00208-006-0043-5
  4. X. Cao, First eigenvalues of geometric operators under the Ricci flow, Proc. Amer. Math. Soc., 136(2008), 4075-4078. https://doi.org/10.1090/S0002-9939-08-09533-6
  5. L. F. D. Cerbo, Eigenvalues of the Laplacian under the Ricci flow, Rend. Mat. Appl. (7), 27(2007), 183-195.
  6. Q.-M. Cheng and H. C. Yang, Estimates on eigenvalues of Laplacian, Math. Ann., 331(2005), 445-460. https://doi.org/10.1007/s00208-004-0589-z
  7. B. Chow, The Yamabe flow on locally conformally flat manifolds with positive Ricci curvature, Comm. Pure Appl. Math., 45(8)(1992), 1003-1014. https://doi.org/10.1002/cpa.3160450805
  8. B. Chow and D. Knopf, The Ricci flow: an introduction, Mathematical Surveys and Monographs 110, AMS, 2004.
  9. S. Fang and F. Yang, First eigenvalues of geometric operators under the Yamabe flow, Bull. Korean Math. Soc., 53(2016), 1113-1122. https://doi.org/10.4134/BKMS.b150530
  10. G. Perelman, The entropy formula for the Ricci flow and its geometric applications, (2002), ArXiv:math/0211159.
  11. L. F. Wang, Eigenvalue estimate for the weighted p-Laplacian, Ann. Mat. Pura Appl. (4), 191(2012), 539-550. https://doi.org/10.1007/s10231-011-0195-0
  12. L. F.Wang, Gradient estimates on the weighted p-Laplace heat eqaution, J. Differential equations, 264(2018), 506-524. https://doi.org/10.1016/j.jde.2017.09.012
  13. J. Y. Wu, First eigenvalue monotonicity for the p-Laplace operator under the Ricci flow, Acta Math. Sin. (Engl. Ser.), 27(8)(2011), 1591-1598. https://doi.org/10.1007/s10114-011-8565-5
  14. F. Zeng, Q. He and B. Chen, Monotonicity of eigenvalues of geometric operators along the Ricci-Bourguignon flow, Pacific J. Math., 296(1)(2018), 1-20. https://doi.org/10.2140/pjm.2018.296.1